This is, essentially, a cyclic model; that is, the Universe goes through repeated "re-birth" phases so had no beginning and will have no end. Rooted in Superstring and M-Theory, it proposes that our Universe is a 3-Brane (3 space and one time dimension) that is embedded in an eleven dimensional "Bulk". One of the extra dimensions is macroscopic, while the remaining six space dimensions are compactified, presumably like M-Theory into a Calabi-Yau manifold. This extra dimension is bounded by "our" brane, with net positive energy, and a second brane with net negative energy. Sometimes, the two branes collide, as the dimension between them contracts, and this causes a new localized expansion phase, starting with a big bang like event having very large, but finite, temperatures and densities. While the two branes are essentially flat, both would have Quantum fluctuations so different parts would collide at very slightly different times leading to the observed flat Universe with small density fluctuations.
From our perspective, the most recent collision happened 13.7 billion years ago triggering the birth of the Universe around us. Each phase could take on the order of trillions of years, and the theory implies that the rate of expansion increases. Over these very long periods, black holes would dissipate through Hawking radiation, and ultimately, one could perceive that there would be less than one particle within each Hubble Volume of space. Dark energy is proposed as a force within the bulk that acts between the branes. They propose that this accelerated expansion, caused by dark energy, dilutes the entropy of the universe so that it returns to its original vacuum state before it begins to contract and "bounce" to begin a new cycle. This neatly gets around the issue of increasing entropy, the second law of thermodynamics, that plagued earlier versions of the cyclic Universe.
M-Theory remains controversial and unproven, and this theory depends critically on Superstrings. In particular, the requirement that most of the particles that we find are represented by open ended strings that are, therefore, rooted in "our" brane, while gravitons, the messenger particles for the force of gravity, are closed like little loops of string. They are free to move in the bulk away from our brane providing an explanation for the incredible weakness of gravity compared to the other forces. So we have speculative cosmology that is based on speculative physics! On the other hand, without speculation, we do not move forwards and this theory certainly explains a number of difficulties inherent in the inflationary big bang; not least of which is cosmic inflation itself, with which this theory dispenses. Certainly, it avoids any singularity as the contraction phase never reaches a complete "crunch", although the dimension between the branes shrinks to zero length, briefly, during the collision.
In its early stages, the Inflationary Big Bang model would have generated a massive amount of gravitational radiation, and eventually, we should be able to detect this as polarization of the cosmic background radiation. There is no equivalent in the brane model. So, if polarization is detected in the microwave background, it would represent further evidence for the Big Bang theory, while if it is not found, something like the Steinhardt-Turok model could be closer to the truth. As of today (March 2011) the jury is still out. Analysis has shown that the current CMB data is not detailed enough to provide proof one way or the other, so we must await a more detailed analysis of the CMB.
This theory fits comfortably with M-Theory, indeed it relies on it, and looks as though it would sit well with something like 5-dimensional warped geometry theory too. It explains both the recently discovered accelerated expansion of the Universe and dark energy more naturally than the standard big bang cosmology. Big pluses are that it removes the singularity that is so awkward in kicking off Big Bang models, as well as removing the need for cosmological inflation. Finally, this model does not produce magnetic monopoles as the temperatures are never high enough to create such massive particles, though there are several ways to explain them away in Big Bang cosmology too.
There are, however, some issues with the model. Certainly, M-Theory/String Theory has no mathematical background to brane collisions, and there are no known, or even proposed, particles to represent the force between the two branes.
The Steinhardt-Turok paper is available on the Science website. You need to have a free subscription to read it, but it only takes a few moments to set up. There is also an article about the theory by Paul Steinhardt that provides some additional information in lay terms.
The Steinhardt-Turok "Brane Collision" or Ekpyrotic Model
Astronomy & Cosmology